Analysis – Counting the rationals

It is easy to see that to show that is countable, it suffices to show the countability of , or even of .

I.

There is a straightforward way of enumerating the latter: First list the fractions with denominator , then those with denominator (skipping those already listed), then those with denominator (again, skipping repetitions), etc. This list begins

Cantor’s first proof of the uncountability of the reals (the nested intervals argument) from 1874 proceeds as follows:

Given any (injective) sequence of reals, we want to exhibit a real that was not listed. There are two cases: Either there are with , and such that there is no with , in which case we are obviously done, or (more interestingly), whenever , we can find an strictly in between (the range of the sequence is dense in itself). Assume we are in this situation.

Define two sequences and as follows:

First, and .

For definiteness, suppose that . The other case is treated similarly. Let be , where is least such that . Then define as , where is least such that .

In general, given , we define as , where is least such that , and then define as , where is least such that . Note that these sequences are well defined, because of our density assumption.

The construction just described ensures that if , then:

For any , we have that , and

The intervals are nested and decreasing:

It follows (from the completeness of the reals) that , and (from 1.) that any real in this intersection is not in the range of the sequence

It turns out that if we carry out Cantor’s construction when the sequence of is the enumeration of the rationals in we began with, then , where is the golden ratio.

The rationals are used to label the nodes of the infinite complete binary tree, and the resulting enumeration simply follows the nodes of the tree, lexicographically.

We begin by putting at the root of the tree. Once a node has been labelled , its left successor is labelled , and its right successor is latex .

And that’s all! The list so produced begins

The proof that this is indeed a bijective listing of is remarkably simple; one verifies in order the following claims:

The numerator and denominator of any of the assigned fractions are relative prime.

Every positive rational is assigned to some node.

Every positive rational is assigned to some node.

For this, one proceeds by induction. For example, if there is a fraction not in reduced form, and used as a label, pick such a fraction appearing in as small a level as possible, and note that the fraction cannot be . A contradiction is now attained by noting that .

Similarly, if appears in more than one node, then , and its immediate predecessor (either or , depending on whether or ) must also appear more than once.

Finally, if some fraction is not listed, we can choose its denominator least among the denominators of all skipped fractions, and then choose its numerator least among the numerators of all skipped fractions with denominator . A contradiction follows because , and if , then must also have been skipped, but , while if , then must have been skipped, but .

This enumeration is due to Neil Calkin and Herbert Wilf, who also showed that it has the following nice combinatorial properties:

There is a sequence of positive integers such that the -th fraction in the enumeration is just (in reduced form). In particular, the denominator of a fraction is the numerator of its successor in the enumeration. So , , , , , , , etc.

In fact, is precisely the number of ways of writing as a sum of powers of , where each power can be used at most twice. For example, , because we can write as , as , or as .

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